Modulated Bi-orthogonal Polynomials on the Unit Circle: The 2j-k and j-2k Systems

2021/06/29 by Roozbeh Gharakhloo, Gharakhloo, Roozbeh, N. S. Witte +1 · 1 citation
Chemistry · Mathematics · #11M50 #33C45 #39A06 #42A52 #42A80 #47B35 #Advanced Algebra and Geometry #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Molecular spectroscopy and chirality

paper · pdf · doi:10.48550/arxiv.2106.15079

openalex publication_date 2021/06/29 · openalex created_date 2023/06/22 · openalex updated_date 2026/07/28

Abstract

We construct the systems of bi-orthogonal polynomials on the unit circle where the Toeplitz structure of the moment determinants is replaced by det(w_2j-k)_{0≤ j,k ≤ N-1} and the corresponding Vandermonde modulus squared is replaced by ∏_{1 ≤ j < k ≤ N}(ζ²_k - ζ²_j)(ζ^-1_k - ζ^-1_j) . This is the simplest case of a general system of pj-qk with p,q co-prime integers. We derive analogues of the structures well known in the Toeplitz case: third order recurrence relations, determinantal and multiple-integral representations, their reproducing kernel and Christoffel-Darboux sum, and associated (Carathéodory) functions. We close by giving full explicit details for the system defined by the simple weight w(ζ)=e^ζ, which is a specialisation of a weight arising from averages of moments of derivatives of characteristic polynomials over USp(2N), SO(2N) and O^-(2N).

Cited by

• A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants II

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